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The Journal of Chemical Physics

Second order many-body perturbation approximations to the coupled cluster Green’s function

Marcel Nooijen, J. G. Snijders

Published Jan 22, 1995• 162 citations
Publisher

Abstract

The time-consuming step in coupled cluster Green’s function or equivalently equation of motion coupled cluster calculations of ionization potentials is the solution of the CCSD equations. We investigate here the accuracy that can be obtained if the CCSD coefficients are replaced by their MBPT(2) analogs. We discuss some additional diagonal approximations that might prove especially useful in polymer calculations, and compare with traditional Green’s function calculations based on a second order approximation to the irreducible self-energy.

DOI
10.1063/1.468900

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The time-consuming step in coupled cluster Green’s function or equivalently equation of motion coupled cluster calculations of ionization potentials is the solution of the CCSD equations. We investigate here the accuracy that can be obtained if the CCSD coefficients are replaced by their MBPT(2) analogs. We discuss some additional diagonal approximations that might prove especially useful in polymer calculations, and compare with traditional Green’s function calculations based on a second order approximation to the irreducible self-energy.
Connection between GWGWGW and Extended Coupled Cluster
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Coupled-cluster (CC) theory and Green’s function many-body perturbation theory (MBPT) have long evolved as distinct yet complementary frameworks for describing electronic correlation. While CC methods employ exponential wave function parametrizations that guarantee size extensivity and systematic improvability, Green’s function approaches, such as the GW approximation, describe quasiparticle and optical excitations through diagrammatic resummations. Recent analyses have established a formal correspondence between these frameworks: the GW approximation is equivalent to an equation-of-motion (EOM) treatment of the direct-ring coupled-cluster doubles (drCCD) method. Within this context, the extended CC (ECC) ansatz offers a unified framework connecting CC and MBPT. This formulation bridges CC-based and Green’s function-based methods, providing novel avenues for incorporating vertex corrections within a CC framework that preserves a sum-over-state representation of the self-energy and leads to potentially systematically improvable Green’s function approaches.
Recently the possible divergence of many-body perturbation theory or many-body green function (propagator) methods with lattice summations in extended systems has been raised. The convergence of these methods with lattice summations is not only the key to establishing their correct scaling properties in inhomogeneous systems, but is also a necessity if numerical calculations are to be meaningful. In this report, it is rigorously shown that many-body perturbation theory (MBPT), coupled cluster theory (CC), and many-body green function (MBGF) methods all converge uniformly with lattice summations, although the integrand for the integration over the reciprocal lattice vector, k, could become infinite at special k values. Our proof is given not only for infinite polymers but also for crystals. In our proof, only the continuity of the zeroth-order band structure and Bloch orbitals with k is used. We show that MBPT, CC, and MBGF methods converge with the radius of the lattice summation range R at least as fast as 1/R1/n where n is the dimensionality of the system. It could be much faster if the zeroth-order band structure and Bloch orbitals have continuous first- or even higher-order derivatives with k. In practical numerical calculations, one should not pursue the convergence of the integrand, but the convergence of the correction considered with lattice summations.
Diagrammatic and Coupled Cluster techniques are used to develop an approach to the single-particle Green's function G which concentrates on G directly rather than first approximating the irreducible self-energy and then solving Dyson's equation. As a consequence the ionization and attachment parts of the Green's function satisfy completely decoupled sets of equations. The proposed Coupled Cluster Green's Function method (CCGF) is intimately connected to both Coupled Cluster Linear Response Theory (CCLRT) and the Normal Coupled Cluster Method (NCCM). These relations are discussed in detail. © 1992 John Wiley & Sons, Inc.
The workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.
In this paper we derive basic properties of the Green's-function matrix elements stemming from the exponential coupled-cluster (CC) parametrization of the ground-state wave function. We demonstrate that all intermediates used to express the retarded (or, equivalently, ionized) part of the Green's function in the $\ensuremath{\omega}$ representation can be expressed only through connected diagrams. Similar properties are also shared by the first-order $\ensuremath{\omega}$ derivative of the retarded part of the CC Green's function. Moreover, the first-order $\ensuremath{\omega}$ derivative of the CC Green's function can be evaluated analytically. This result can be generalized to any order of $\ensuremath{\omega}$ derivatives. Through the Dyson equation, derivatives of the corresponding CC self-energy operator can be evaluated analytically. In analogy to the CC Green's function, the corresponding CC self-energy operator can be represented by connected terms. Our analysis can easily be generalized to the advanced part of the CC Green's function.